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BSCross product of standard unit vector in Cartesian coordinate system.
@bsumathdept
#Cartesian Coordinate System
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EXCoordinate Systems | Cartesian and Polar
Explore the fundamental connection between Cartesian and polar coordinates in this short video about coordinate systems. Discover how Cartesian coordinates, defined by horizontal (x) and vertical (y) distances, seamlessly relate to polar coordinates, represented by distance (r) from the origin and angle (θ) with respect to the horizontal axis.
Polar coordinates offer a different perspective on describing points in a plane compared to Cartesian coordinates. They’re particularly useful in scenarios where circular or rotational symmetry is involved, such as in physics, engineering, and mathematics.
One significant advantage of polar coordinates is their ability to simplify equations, especially those involving circular or radial patterns. For example, equations describing curves or shapes with rotational symmetry, like circles, ellipses, spirals, and sectors of circles, are often more straightforward in polar form.
#mathematics #math #maths
@explain.stuff
402.2K
EESpherical vs Cartesian coordinate systems
@eeanimation
73.8K
STThis elegant mathematical visualization video showcases six different mathematical functions being graphed in real-time on Cartesian coordinate systems, demonstrating the beautiful patterns that emerge from mathematical equations. And these equations are turned into sounds.
👉Follow @star.ai.media for more amazing tech, and AI insights.
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Links in bio and stories!👆
@star.ai.media
214.1K
MAPolar and Cartesian coordinates are two different ways to describe the location of a point in the plane. In the Cartesian system, a point is identified by an ordered pair (x, y), which tells us how far the point is from the origin horizontally and vertically. The x coordinate measures movement along the horizontal axis, while the y coordinate measures movement along the vertical axis. This system is especially convenient for describing lines, parabolas, and other curves defined by equations such as y equals x squared or x equals 3. Because distances are measured in straight horizontal and vertical directions, Cartesian coordinates are often the most natural choice when working with algebraic equations and rectangular grids.
In contrast, polar coordinates describe a point using a distance and an angle. Instead of (x, y), we write a point as (r, theta), where r is the distance from the origin and theta is the angle measured from the positive x axis. This system is particularly useful for curves that involve rotation or symmetry around a central point, such as circles, spirals, and rose curves. A single geometric object can often be expressed in both systems, and there are formulas that connect them: x equals r cosine theta and y equals r sine theta. By switching between polar and Cartesian coordinates, we can choose the system that makes a problem simpler or reveals hidden structure in a curve.
Like this video and follow @mathswithmuza for more!
#math #algebra #graphs #calculus #draw
@mathswithmuza
46.7K
THSpherical (r, φ, θ) to Cartesian (x, y, z):
x = r * cos(θ) * sin(φ)
y = r * sin(θ) * sin(φ)
z = r * cos(φ)
Cartesian (x, y, z) to Spherical (r, φ, θ):
r = √(x² + y² + z²)
φ = arccos(z / r)
θ = arctan(y / x)
These are spherical coordinates. They were used in part of this animation to calculate the spheres. Spherical coordinates define locations in space with three values. The distance from the center, called the radius, denoted by the letter r, the angle away from the north pole, or polar or zenith angle, denoted by the letter φ (“phi”), and the angle around the pole, call the azimuthal angle, denoted by the letter θ (“theta”). These three quantities can define any location in three-dimensional space.
Alternatively, rectilinear coordinates, or Cartesian coordinates, also define any location in three-dimensional space, but instead use the three quantities of x, y, and z, the distances in the right-left, forward-backwards, and up-down directions.
The equations above shows the relationship between the two coordinate systems, that is, if you have the three values from the one system, you can use these equations to find the three values in the other system.
If we are to truly awaken genius, we must see these equations as descriptions of a self-evident mathematical truth. Not simply memorize them, but learn to see them as descriptions of something we know. Can you see it?
This animation was created with the Java programming language in the Processing IDE (interactive development environment). Below you can see a snippet of code that uses spherical coordinates to generate the frames in this animation.
float theta = map(i, 0, n, 0, TAU);
float thetaStep = TAU / n;
float x = r * sin(phi) * cos(theta);
float y = r * sin(phi) * sin(theta);
float z = r * cos(phi);
AVector v = new AVector(x, y, z);
v.rotateY(rotatePhi);
v.rotateZ(rotateTheta);
#SphericalCoordinates #Mathematics #Enlightenment #Meditation #Education #Spirituality #Coding #ComputerScience #Love #Bhakti
@thebhaktimathguru
3.4K
DYPolar and Cartesian coordinates are two different ways to describe the location of a point in the plane. In the Cartesian system, a point is identified by an ordered pair (x, y), which tells us how far the point is from the origin horizontally and vertically. The x coordinate measures movement along the horizontal axis, while the y coordinate measures movement along the vertical axis. This system is especially convenient for describing lines, parabolas, and other curves defined by equations such as y equals x squared or x equals 3. Because distances are measured in straight horizontal and vertical directions, Cartesian coordinates are often the most natural choice when working with algebraic equations and rectangular grids.
In contrast, polar coordinates describe a point using a distance and an angle. Instead of (x, y), we write a point as (r, theta), where r is the distance from the origin and theta is the angle measured from the positive x axis. This system is particularly useful for curves that involve rotation or symmetry around a central point, such as circles, spirals, and rose curves. A single geometric object can often be expressed in both systems, and there are formulas that connect them: x equals r cosine theta and y equals r sine theta. By switching between polar and Cartesian coordinates, we can choose the system that makes a problem simpler or reveals hidden structure in a curve.
#math #algebra #graphs #calculus #draw
@dybydx.ai
1.5M
MAEnjoy some more polar and cartesian comparisons and follow @mathswithmuza for more! #math #mathstudent #learn #college #school #teach #foryou #fyp #curious #how #why #chatgpt #ai #coding #programming #study #exam #animation #visual
@mathswithmuza
696.2K
TETranslating rotation into waves.
The unit circle is the engine of trigonometry. As the angle rotates, the Sine, Cosine, and Tangent curves emerge perfectly in sync.
Math isn't just calculated; it is seen.
■ Q.E.D.
#trigonometry #math #mathvisuals
@tensor.qed
743.1K
NAThat's really cool!!! #mathematics #physics #science #physicsfun #sci
@narendra_1.1.1.1.1
14.9M
MATrigonometry becomes much clearer when we connect algebraic formulas to geometry. This visual explanation starts with the unit circle, a circle of radius 1 centered at the origin, which forms the foundation of trigonometric definitions. Any angle measured from the positive x-axis determines a point on the unit circle, and the coordinates of this point directly define cosine and sine. The x-coordinate represents cos θ, while the y-coordinate represents sin θ.
By dropping a perpendicular from this point to the x-axis, we naturally obtain a right-angled triangle. This links the unit circle to the familiar concepts of adjacent, opposite, and hypotenuse. From this construction, tan θ appears as the ratio of sine to cosine and is visualized using the tangent line to the unit circle.
The animation then extends these ideas to graphs. As the angle increases, the changing sine, cosine, and tangent values trace smooth curves, forming the sin graph, cos graph, and tan graph. This shows how circular motion generates periodic wave patterns.
This approach preserves correct mathematical meaning while helping students see how angles, triangles, and graphs are deeply connected—making trigonometry logical, visual, and memorable.
#math #trigonometry #fyp #trending
@mathematisa
228
CO📍 "Specifying Location in 3D Space... VISUALLY!"
In 2D, you move right & up.
But in 3D? You go right, forward, and UP!
This is how we locate anything in space using the Cartesian Coordinate System:
➤ X-axis → Left & Right
➤ Y-axis → Forward & Back
➤ Z-axis → Up & Down
From robotics to 3D graphics, every object, drone, or spaceship uses this exact system to find its position 🌍🚁
🟡 A(2,2,3)
🟠 B(4,-1,2)
Arrows show how we move step by step through X → Y → Z
🎥 And yes… we added a smooth fly-around camera to bring it to life 👀
Hit 🔁 if your mind just upgraded from 2D → 3D 😏
#3dcoordinates #manim #mathvisuals #coordinategeometry #cartesiancoordinates #vectorspace #mathanimation #mathsxmanim #learnvisually #3dmath #code2model
@code2model
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